Mean and Variance of Extreme Value Distributions

Computes the mean and variance of common extreme value distributions given their parameters.

Usage

mgumbel(loc = 0, scale = 1)

mrevgumbel(loc = 0, scale = 1)

mfrechet(loc = 0, scale = 1, shape = 1)

mrevweibull(loc = 0, scale = 1, shape = 1)

mgev(loc = 0, scale = 1, shape = 0)

mgpd(loc = 0, scale = 1, shape = 0)

mgompertz(shape, rate = 1)

Arguments

loc

location parameter.

scale

scale parameter.

shape, rate

shape and rate parameters.

Value

A named numeric vector with elements mean and variance. Returns NA where moments do not exist.

Details

Distribution	Mean	Variance
Gumbel	\(a + b\gamma\)	\(\frac{\pi^2}{6}b^2\)
Reverse Gumbel	\(a - b\gamma\)	\(\frac{\pi^2}{6}b^2\)
Fréchet	\(a + b\Gamma(1 - 1/s) \quad (s > 1)\)	\(b^2\left[\Gamma(1 - 2/s) - \Gamma(1 - 1/s)^2\right] \quad (s > 2)\)
Reverse Weibull	\(a - b\Gamma(1 + 1/s)\)	\(b^2\left[\Gamma(1 + 2/s) - \Gamma(1 + 1/s)^2\right]\)
GEV	\(a + b\frac{\Gamma(1-s)-1}{s} \quad (s \ne 0,\ s < 1)\)	\(b^2\frac{\Gamma(1-2s)-\Gamma(1-s)^2}{s^2} \quad (s \ne 0,\ s < 1/2)\)
GPD	\(a + \frac{b}{1-s} \quad (s < 1)\)	\(\frac{b^2}{(1-s)^2(1-2s)} \quad (s < 1/2)\)
Gompertz	numerical integration for \(\alpha > 0\)	dito
	\(1/\beta\) for \(\alpha = 0\);	\(1/\beta^2\) for \(\alpha = 0\);
	NA for \(\alpha < 0\)	dito

For the first six distributions, \(a\) = loc, \(b\) = scale, and \(s\) = shape. For the GEV with \(s = 0\), the Gumbel moments apply. Furthermore, \(\gamma \approx 0.5772\) is the Euler-Mascheroni constant. For the Gompertz distribution, \(\alpha\) = shape and \(\beta\) = rate; moments for \(\alpha > 0\) are computed numerically by integration.

References

Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. Springer.

Kotz, S. and Nadarajah, S. (2000) Extreme Value Distributions. Imperial College Press.

See also

dgumbel(), drevgumbel(), dfrechet(), drevweibull(), dgev(), dgpd(), distributions-overview

Examples

mgumbel(loc = 0, scale = 1)
#>      mean  variance 
#> 0.5772157 1.6449341 
mrevgumbel(loc = 0, scale = 1)
#>       mean   variance 
#> -0.5772157  1.6449341 
mfrechet(loc = 0, scale = 1, shape = 3)
#>      mean  variance 
#> 1.3541179 0.8453031 
mrevweibull(loc = 0, scale = 1, shape = 2)
#>       mean   variance 
#> -0.8862269  0.2146018 
mgev(loc = 0, scale = 1, shape = 0)
#>      mean  variance 
#> 0.5772157 1.6449341 
mgev(loc = 0, scale = 1, shape = 0.3)
#>      mean  variance 
#> 0.9935178 5.9245766 
mgev(loc = 0, scale = 1, shape = -0.3)
#>      mean  variance 
#> 0.3417643 0.9784633 
mgpd(loc = 0, scale = 1, shape = 0.3)
#>     mean variance 
#> 1.428571 5.102041